Higher-order Schrodinger operators with singular potentials (Nonlinear evolution equations and mathematical modeling)

Higher-order

Schr\"odinger

operators

with singular

potentials

東京理科大学・理 岡沢 登 (Noboru Okazawa)

東京理科大学・理D1 田村 博志 (Hiroshi Tamura)

東京理科大学・理 横田 智巳 (Tomomi Yokota)

Department _{of Mathematics, Science University of Tokyo}

Abstract. Theselfadjointness of$\Delta^{2}+\kappa|x|^{-4}(\kappa\in \mathbb{R})$in $L^{2}(\mathbb{R}^{N})$ and the7Yl-accretivity of

$\Delta^{2}+\kappa|x|^{-4}(\kappa\in \mathbb{C})$ in $L^{2}(\mathbb{R}^{N})$ are established _ais applications of perturbation theorems

for nonnegative selfadjoint operators. The key lies in two new inequalities derived by

using two real orcomplex parameters.

1.

Introduction

and

results

Let $N\in \mathbb{N}$

.

Then this paper is concerned with the selfadjointness of_{$\Delta^{2}+\kappa|x|^{-4}$} (when

$\kappa\in \mathbb{R})$, and the m-accretivity of$\Delta^{2}+\kappa|x|^{-4}$ (when $\kappa\in \mathbb{C}$) in the (complex) Hilbert space

$L^{2}(\mathbb{R}^{N})$

.

Here $\Delta^{2}$and

$|x|^{-4}$ arenonnegative selfadjoint operators in $L^{2}(\mathbb{R}^{N})$, with domains

$D(\Delta^{2});=H^{4}(\mathbb{R}^{N})$ and $D(|x|^{-4});=\{u\in L^{2}(\mathbb{R}^{N});|x|^{-4}u\in L^{2}(\mathbb{R}^{N})\}$, respectively.

First we consider the selfadjointness of $\Delta^{2}+\kappa|x|^{-4}(\kappa\in \mathbb{R})$

.

On the one hand, it is

worth noticing that the relation betweensimpleroperators $-\Delta$ and $|x|^{\sim 2}$ is already known

as a model

case.

In [8] it ha.$s$ been proved that $-\Delta+t|x|^{-2}$ is m-accretive in $L^{p}(\mathbb{R}^{N})$ for

$t>a_{0}(p)$ and $-\Delta+a_{0}(p)|x|^{-2}$ is essentially m-accretive in $L^{p}(\mathbb{R}^{N})(1<p<\infty)$, where $a_{0}(p)$ is defined as

$a_{0}(p):=\{\begin{array}{ll}p^{-2}(p-1)(2p-N)N, 2(1-N^{-1})\leq p<\infty,-p^{-2}(p-1)(N-2)^{2}, 1<p<2(1-N^{-1}).\end{array}$

In particular, if $p=2$, then $a_{0}(2)=4^{-1}(4-N)N$ _{and m-accretivity is replaced with}

nonnegative selfadjointness. A proofofthe selfadjointness in [7] isbased on the inequality

${\rm Re}(-\Delta u, (|x|^{2}+n^{-1})^{-1}u)\geq-a_{0}(2)\Vert(|x|^{2}+n^{-1})^{-1}u\Vert^{2}$, $u\in H^{2}(\mathbb{R}^{N})$,

where $(|x|^{2}+n^{-1})^{-1}=|x|^{-2}(1+7|^{-1}|x|^{-2})^{-1}$ is the Yosida approximation of $|x|^{-2}(n\in \mathbb{N})$

.

On the other hand, there

seems

to be few works about the selfadjointness of higher order

elliptic operators. In [6] Nguyen discussed the selfadjointness of general

even

order elliptic

operators under several assumptions. However, his result cannot be applied to determine

the critical bound of $\kappa$ for the selfadjointness of $\Delta^{2}+\kappa|x|^{-4}$

.

The first purpose of this paper is to establish the following

Theorem 1.1. Put $A:=\Delta^{2}$ and _{$B:=|x|^{-4}$}

.

Let $\kappa_{0}(N)$ be

defined

as

Then the following (i) and (ii) hold.

(i)

_If

$N\leq 8$, then $B$ is $(A+\kappa B)$-bounded

for

$\kappa>\kappa_{0}(N)o_{*}s$

$\Vert Bu\Vert\leq(\kappa-\kappa_{0}(N))^{-1}\Vert(A+\kappa B)u\Vert$, $u\in D(A+\kappa B):=D(A)\cap D(B)$,

and $A+\kappa B$ is nonnegative selfadjoint

for

$\kappa>\kappa_{0}(N)$

.

Moreover, $A+\kappa_{0}(N)B$ is

nonneg-ative and essentially selfadjoint.

(ii)

_If

$N\geq 9_{f}$ then $B$ is A-bounded

as

(1.2) $\Vert B\uparrow\nu\Vert\leq\frac{16}{N(N-8)(N^{2}-16)}\Vert Au\Vert$ , $u\in D(A)\subset D(B)$,

and $A+\kappa B$ is nonnegative selfadjoint

for

$\kappa>\kappa_{0}(N)$

.

Moreover, $A+\kappa_{0}(N)B$ is

nonneg-ative and essentially selfadjoint in $L^{2}(\mathbb{R}^{N})$

.

Next

we

shall find $\Omega\subset \mathbb{C}$ such that $\{\Delta^{2}+\kappa|x|^{-4};\kappa\in\Omega\}$ is a holomorphic family of

type (A) in the sense of Kato [4, Section VII.2]. We review it in a simple case.

Definition 1. Let $X$ be $a$

oeflexive

complex Banach space. Let $\Omega$ be

a

domain in $\mathbb{C}$ and

$\{T(\kappa);\kappa\in\Omega\}$

a

family

of

linear operators in X. Then $\{T(\kappa);\kappa\in\Omega\}$ is said to be

a

holomorphic family

_of

type $(A)$

if

(i) $T(\kappa)$ is closed in $X$ and _{$D(T(\kappa))=D$} independent

of

$\kappa$;

(ii) $\kappa\mapsto T(\kappa)u$ is holomorphic in $\Omega$

for

every $u\in D$

.

Kato [5] proved that $\{-\Delta+\kappa|x|^{-2};\kappa\in\Omega_{1}\}$ forms

a

holomorphic family oftype (A)

in $L^{2}(\mathbb{R}^{N})$, where

$\Omega_{1}:=\{\xi+i\eta\in \mathbb{C};\eta^{2}>4(\beta-\xi)\}$, $\beta:=(N-2)^{2}/4$

.

Borisov-Okazawa [1] proved that $\{d/dx+\kappa|x|^{-1};\kappa\in\Omega_{2}\}$ forms a holomorphic family of

type (A) in $L^{p}(0, \infty)(1<p<\infty)$, where

$\Omega_{2}:=\{\kappa\in \mathbb{C};{\rm Re}\kappa>-\frac{1}{p}\}$ , $p^{-1}+p^{\prime-1}=1$

.

In both $ca_{\wedge}ses$ it is essential to find $\Sigma_{j}$ $:=\Omega_{j^{\mathbb{C}}}$, the complement of$\Omega_{j}(j=1,2)$. Concerning

forth order elliptic operators, there seems to be no preceding work on holomorphic family

of type (A). So we clarify the region where $\Delta^{2}+\kappa|x|^{-4}$ forms a holomorphic family of

type (A) and where $\Delta^{2}+\kappa|x|^{-4}$ is m-accretive in $L^{2}(\mathbb{R}^{N})$ (the definition of (regular)

m-accretivity will be given in Section 3). Our second result here is stated

as

follows.

Theorem 1.2. Let $A$ and $B$ be the

same as

in Theorem 1.1. Let $\Sigma$ be a closed

convex

subset

_of

$\mathbb{C}$ (see Figure 1) such that

$\Sigma:=\{\xi+i\eta\in \mathbb{C};\xi\leq k_{1},$ $\eta^{2}\leq 64(\sqrt{k_{1}-\xi}+(10+N-\frac{N^{2}}{4}))(\sqrt{k_{1}-\xi}+8)^{2}\}$,

where the constant $k_{1}$ is

defined

in (1.1); replace $\Sigma$ with

if

$N\geq 9$ [the constant $k_{2}$ is also

defined

in (1.1)]. Then the following (i) and (ii) hold.

(i) $B$ i,s $(A+\kappa B)$-bounded

for

$\kappa\in\Sigma^{c}$, with

$\Vert Bu\Vert\leq$ dist$(\kappa, \Sigma)^{-1}\Vert(A+\kappa B)u\Vert$, $u\in D(A)\cap D(B)$,

and $\{A+\kappa B;\kappa\in\Sigma^{c}\}$

forms

a holomorphic family

of

type (A) in $L^{2}(\mathbb{R}^{N})$

.

(ii) $A+\kappa B$ is m-accretive

on

_{$D(A)\cap D(B)$}

for

$\kappa\in\Sigma^{c}$ with ${\rm Re}\kappa\geq-\alpha_{0},$ $A+\kappa B$

is regularly m-accretive on $D(A)\cap D(B)$

for

$\kappa\in\Sigma^{c}$ with ${\rm Re}\kappa>-\alpha_{0}$ and $A+\kappa B$ is

essentially m-accretive in $L^{2}(\mathbb{R}^{N})$

for

$\kappa\in\partial\Sigma$ with ${\rm Re}\kappa\geq-\alpha_{0}$, where $\alpha_{0}$ is

defined

as

(1.3) $\alpha_{0}:=\{$ $\frac{0N^{2}}{16}(N-4)^{2}$

, $N\geq 5$

.

$N\leq 4$,

In particular,

_if

$\kappa\in \mathbb{R}$, then m-accretivity can be replaced with nonnegative selfadjointness.

Figure 1: The images of $\Sigma$ for $N=4,5,8,9$ and the value of

$-\alpha_{0}$

The constant $\alpha_{0}$ in (1.3) appears in the Rellich inequality

In [3] Davies-Hinz have shown Hardy or Rellich type inequalities between $(-\triangle)^{m}$ and $|x|^{-2m}(m\in \mathbb{N})$, and it helps us to construct the theory of the selfadjointness.

In Section 2 we review abstract theorems ba.sed on [8]. In Section 3 we prepare

abstract theorems ba.sed on Kato [5] (however, the$a_{\wedge}ss\iota imption$ and conclusions

are

slightly

changed). In Section 4 we derive some new inequalities by using two real parameters and

prove Theorem 1.1 by applying abstract theorems prepared in Section 2. In Section 5 we

generalize inequalities obtained in Section 4 by using two complex parameters and prove

Theorem 1.2 by applying abstract theorenis prepared in Section 3.

2.

Perturbation theory toward Theorem

1.1

This section is a short review of the perturbation theory developed in [7] and [8] for

m-accretive operators in a Banach space. The following two theorems

are

the special

cases of those in [8].

Theorem 2.1 ([8, Theorem 1.6]). Let $A$ and $B$ be nonnegative selfadjoint operators in

a

Hilbert space H. Let $B_{\epsilon}$ $:=B(1+\epsilon B)^{-1}$ be the Yosida approximation

of

B. Assume

that there exists some $k_{0}\geq 0$ such that

(2.1) ${\rm Re}(Au, B_{\epsilon}u)\geq-k_{0}\Vert B_{\epsilon}u||^{2}$, _{$u\in D(A)$}.

Then $B$ is $(A+kB)$-bounded

for

$k>k_{0}$

as

(2.2) $\Vert Bu\Vert\leq(k-k_{0})^{-1}\Vert(A+kB)u\Vert$, $u\in D(A+kB)$,

and hence $A+kB$ is closed in $H$

for

_{$k>k_{0}$}

.

Moreover, $A+kB$ is nonnegative selfadjoint

on

$D(A)\cap D(B)$

for

$k>k_{0}\geq 0$ and $A+k_{0}B$ is nonnegative and essentially selfadjoint

in $H$

.

Theorem 2.2 ([8, Theorem 1.7]). Let $A,$ $B$ and $B_{\epsilon}$ be the

same

as those in Theorem

2.1. Assume that there exists

some

$m_{1}>0$ such that

(2.3) ${\rm Re}(Au, B_{\epsilon}u)\geq m_{1}\Vert B_{\epsilon}u\Vert^{2}$, _{$u\in D(A)$}.

Then $B$ is A-bounded as

(2.4) $\Vert Bu\Vert\leq m_{1}^{-1}\Vert Au\Vert$, _{$u\in D(A)\subset D(B)$},

and $A+kB$ _{is closed} in $H$

for

_{$k>-m_{1}$}. Assume

further

that there erists

some

$m_{2}\geq\sqrt{m_{1}}$

such that $m_{2}^{2}(B_{\epsilon}u, u)\leq(Au, u),$ _{$u\in D(A)$},

or

equivalently

(2.5) $m_{2}\Vert B^{1\prime 2}(1+\epsilon B)^{-1\prime 2}v\Vert\leq\Vert A^{1\prime 2}v\Vert$, $v\in D(A^{1\prime 2})$

Then $A+kB$ is nonnegative selfadjoint in $H$

for

_{$k>-k_{1z}$} and _{$A-k_{1}B$} is nonnegative

3. Perturbation

theory

toward

Theorem 1.2

First we review some definitions required to state Theorems 3.1 and 3.5. Let $A$ be a

linear operator with domain $D(A)$ and range $R(A)$ in a (complex) Hilbert space $H$. Then

$A$ is said to be accretive if ${\rm Re}$(Au,

$u$) $\geq 0$ for every $\dot{\tau}\iota\in D(A)$. An accretive operator $A$ is

said to be m-accretive if $R(A+1)=H$. An m-accretive operator $A$ is said to be regularly

m-accretive if $A$ satisfies for some $\omega\in[0, \pi/2)$ that

$|{\rm Im}$(Au,$u$)$|\leq(\tan\omega){\rm Re}$ (Au,$u$), $u\in D(A)$.

Let $A$ be m-accretive in $H$. Then _{$R(A+\lambda)=H$} holds, with

$\Vert(A+\lambda)^{-1}\Vert\leq({\rm Re}\lambda)^{-1}$ $\forall\lambda\in \mathbb{C}$ with ${\rm Re}\lambda>0$.

Therefore we can define the Yosida approximation $\{A_{\epsilon};\epsilon>0\}$ of $A$:

$A_{\epsilon}:=A(1+\epsilon A)^{-1}$

A nonnegative selfadjoint operator is a typical example of m-accretive operator, while a

symmetric m-accretive operator is nonnegative and selfadjoint (see Br\’ezis [2, Proposition

$V\mathbb{I}.6]$ or Kato [4, Problem V.3.32]$)$.

Next we consider the m-accretivity of$A+\kappa B(\kappa\in \mathbb{C})$ where $A$ and $B$ are nonnegative

selfadjoint operators. Since m-accretive operators are closed and densely defined, we will

first find the set of $\kappa\in \mathbb{C}$ where $A+\kappa B$ is closed (and densely defined). Hence we can

connect the two notions of m-accretivity and holomorphic family of closed operators.

Theorem 3.1. Let $A$ and $B$ be nonnegative selfadjoint operators in H. Let $\Sigma\subset \mathbb{C}_{f}$ and

$\gamma$ : $\mathbb{R}arrow \mathbb{R}$. Assume that $\Sigma$ and $\gamma$ satisfy $(\gamma 1)-(\gamma 4)$ and $(\gamma 5)_{0}$ :

$(\gamma 1)\gamma$ is continuous _$and-\gamma$ is convex,

$(\gamma 2)\gamma(\eta)=\gamma(-\eta)$

for

$\eta\in \mathbb{R}$, $(\gamma 3)\Sigma=\{\xi+i\eta\in \mathbb{C};\xi\leq\gamma(\eta)\}$,

$(\gamma 4)-(Au, B_{\epsilon}u)\in\Sigma$

for

_{$u\in D(A),$} $\Vert B_{\epsilon}u\Vert=1$

for

any $\epsilon>0$,

$(\gamma 5)_{0}0\leq\gamma(0)\Leftrightarrow 0\in\Sigma$

.

Then the following (i) and (ii) hold.

(i) $B$ is $(A+\kappa B)$-bounded

for

$\kappa\in\Sigma^{c}$, with

(3.1)

11

$Bu||\leq$ dist$(\kappa, \Sigma)^{-1}\Vert(A+\kappa B)u\Vert$, $u\in D(A)\cap D(B)$,

and $\{A+\kappa B;\kappa\in\Sigma^{c}\}$

forms

a

holomorphicfamily

of

type (A).

(ii) $A+\kappa B$ is m-accretive in $H$

for

$\kappa\in\Sigma^{c}$ with ${\rm Re}\kappa\geq 0,$ $A+\kappa B$ is regularly

m-accretive in $H$

for

$\kappa\in\Sigma^{c}$ with ${\rm Re}\kappa>0$ and $A+\kappa B$ is essentially m-accretive in $H$

for

$\kappa\in\partial\Sigma$ with ${\rm Re}\kappa\geq 0$.

The proofof Theorem 3.1 is divided into several lemmas.

Proof. Let $\kappa\in\Sigma^{c}$ and $\epsilon>0$. To prove (3.1) we shall show that

(3.2) $\Vert B_{\epsilon}u\Vert\leq$ dist $(\kappa, \Sigma)^{-1}\Vert(A+\kappa B_{\epsilon})u\Vert$ , $u\in D(A)$.

Here we may assume that $B_{\epsilon}u=B(1+\epsilon B)^{-1}u\neq 0$. Setting $v$ $:=\Vert B_{\epsilon}u\Vert^{-1}u$, we see that

$v\in D(A)$ and $\Vert B_{\epsilon}v\Vert=1$

.

it then follows from $(\gamma 4)$ that $-(Av, B_{\epsilon}v)\in\Sigma$

.

Since $\Sigma$ is

closed and

convex

by $(\gamma 1)$, we have

$0<$ dist $(\kappa, \Sigma)\leq|\kappa+(Av, B_{\epsilon}v)|=\Vert B_{\epsilon}\uparrow 4\Vert^{-2}|((A+\kappa B_{\epsilon})u, B_{\epsilon}u)|$,

and hence $\Vert B_{\epsilon}u\Vert^{2}\leq$ dist $(\kappa, \Sigma)^{-1}|((A+\kappa B_{\epsilon})u, B_{\epsilon}u)|$. Now the Cauchy-Schwarz inequality

applies to give (3.2). Letting $\epsilon\downarrow 0$ in (3.2) with $u\in D(A)\cap D(B)$ yields (3.1). The

closedness of $A+\kappa B$ is a consequence of (3.1). This completes the proofof (i) in Theorem

3.1 口

Lemma 3.3. $A+\kappa B$ is m-accretive in $H$

for

$\kappa\in\Sigma^{c}$ with ${\rm Re}\kappa\geq 0$. In particular,

if

${\rm Re}\kappa>0$, then $A+\kappa B$ is regularly m-accretive in $H$, with

(3.3) $|{\rm Im}((A+\kappa B)u, u)|\leq(\tan|\arg\kappa|){\rm Re}((A+\kappa B)u, u)$, $u\in D(A)\cap D(B)$

.

Proof. Since the sum of accretive operators is also accretive, it suffices to show that

(3.4) $R(A+\kappa B+\lambda)=H$, $\lambda>0$

for $\kappa\in\Sigma^{c}$ with ${\rm Re}\kappa\geq 0$. Since $A+\kappa B_{\epsilon}$ is also m-accretive (see [10, Corollary 3.3.3]),

for $f\in H$ and $\epsilon>0$ there exists a unique solution $u_{\epsilon}\in D(A)$ of approximate equation

(3.5) $Au_{\epsilon}+\kappa B_{\epsilon}u_{\epsilon}+\lambda u_{\epsilon}=f$,

satisfying $\Vert u_{\epsilon}||\leq\lambda^{-1}\Vert f\Vert$ and hence $\Vert Au_{\epsilon}+\kappa B_{\epsilon}u_{\epsilon}$

II

$=\Vert f-\lambda u_{\epsilon}$

II

$\leq 2||f||$

.

Therefore we

see from (3.2) that

$\Vert B_{\epsilon}u_{\epsilon}\Vert\leq 2$ dist $(\kappa, \Sigma)^{-1}\Vert f\Vert$,

and hence $\Vert Au_{\epsilon}\Vert\leq 2(1+|\kappa|$dist$(\kappa,$ $\Sigma)^{-1})\Vert f\Vert$. Thus $\Vert u_{\epsilon}\Vert,$ $\Vert$Au,

11

and $\Vert B_{\epsilon}u_{\epsilon}\Vert$

are

bounded

a.s $\epsilon$ tends to zero. This implies that there exist convergent subsequences $\{u_{\epsilon_{n}}\},$ $\{Au_{\epsilon_{n}}\}$

and $\{B_{\epsilon_{n}}u_{\epsilon_{\mathfrak{n}}}\}=\{B(1+\epsilon_{n}B)^{-1}u_{\epsilon_{n}}\}$ for some null sequence $\{\epsilon_{n}\}$. Since $A$ and $B$

are

(weakly) closed, there exists $u:=w-$lini$narrow\infty^{u}\epsilon_{n}\in D(A)\cap D(B)$ such that

$Au_{\epsilon_{n}}arrow Au$ and $B_{\epsilon_{\mathfrak{n}}}u_{\epsilon_{n}}arrow Bu(narrow\infty)$ weakly;

note that $u_{\epsilon}-(1+\epsilon B)^{-1}u_{\epsilon}=\epsilon B_{\epsilon}u_{\epsilon}$

.

Letting $narrow\infty$ in (3.5) with $\epsilon=\epsilon_{n}$ in the weak

topology of $H$, we obtain (3.4). The regular m-accretivity of $A+\kappa B$ for $\kappa\in\Sigma^{c}$ with

${\rm Re}\kappa>0$ follows to consider the numerical range of $A+\kappa B$; $((A+\kappa B)u, u)=||A^{1/2}u||^{2}+\kappa||B^{1/2}u||^{2}$

$\in\{a+\kappa b\in \mathbb{C};a\geq 0, b\geq 0\}$

$\subset\{z\in \mathbb{C};|\arg z|\leq|\arg\kappa|\}$, $u\in D(A)\cap D(B)$.

Lemma 3.4. The closure

_of

$A+\kappa B$ is m-accretive in $H$

for

$\kappa\in\partial\Sigma$ with ${\rm Re}\kappa\geq 0$.

Proof. Let $\kappa\in\partial\Sigma$ with ${\rm Re}\kappa\geq 0$. First we note that $A+\kappa B$ is closable and its

closure is also accretive (cf. [10, Theorem 1.4.5]). Now $(\gamma 1)$ means that there exists

some

(not unique in general) unit outward normal vector $\nu$ of $\partial\Sigma$ at $\kappa$

.

This implies that

$\kappa+t\nu\in\Sigma^{c}(t>0)$, with the properties:

${\rm Re}(\kappa+t\nu)\geq 0$, dist $(\kappa+t\nu, \Sigma)=t$, $t>0$.

This implies that $A+\kappa B(\kappa\in\partial\Sigma)$ is approximated by $A+(\kappa+\nu n)B(\kappa+\nu n\in\Sigma^{c})$

with $n\in \mathbb{N}$. Since ${\rm Re}\kappa+\nu\prime n\geq 0$, we

see

that _{$A+(\kappa+(\nu/n))B$} is m-accretive (see

Lemma 3.3), that is, $f\in H$ there exists a unique solution $u_{n}\in D(A)\cap D(B)$ of

(3.6) $(A+\kappa B)u_{n}+(\nu/n)Bu_{n}+\lambda u_{n}=Au_{n}+(\kappa+(\nu n))B\tau x_{n}+\lambda u_{n}=f$,

satisfying

(3.7) $\Vert u_{n}\Vert\leq\lambda^{-1}\Vert f\Vert$

.

Now we can prove that

11

$(\nu/n)B?4_{n}\Vert=n^{-1}\Vert Bu_{n}$

Il

$\leq 2\Vert f\Vert$. In fact, it follows from (3.1)

that

$\Vert Bu_{n}\Vert\leq$ dist$(\kappa+n^{-1}\nu, \Sigma)^{-1}\Vert(A+(\kappa+\nu n)B)\tau x_{n}\Vert=n\Vert f-\lambda u_{n}\Vert$

$\leq 2n\Vert f\Vert$

.

This yields together with (3.6) that

(3.8) $\Vert(A+\kappa B)u_{n}\Vert\leq 4\Vert f\Vert$ $\forall n\in \mathbb{N}$

To finish the proofwe show that $(\nu\prime n)Bu_{n}$ converges to

zero

weakly in $H$

.

It follows from

(3.7) that for every $v\in D(B)$,

$|((\nu n)Bu_{n}, v)|=n^{-1}|(u_{n}, Bv)|\leq n^{-1}\lambda^{-1}\Vert f\Vert\cdot\Vert Bv\Vertarrow 0,$ _{$narrow\infty$}

.

Since $D(B)$ is dense in $H$ and $n^{-1}\Vert Br4_{t}n\Vert$ is bounded, we can conclude that $n^{-1}B\tau x_{n}arrow 0$

weakly $a_{\iota}snarrow\infty$. Now let $\{u_{n_{k}}\}$ be a convergent subsequence of $\{u_{n}\}$ and put $u:=w-$ $\lim_{karrow\infty}u_{n}k$. Then we have

$(A+\kappa B)u_{n_{k}}=f-\lambda?4_{n_{k^{-}}}(\nu/n)Bu_{n_{k}}$ $arrow f-\lambda u(karrow\infty)$ weakly.

It follows from the (weak) closedness that $u\in D((A+\kappa B)^{\sim})$ and $(A+\kappa B)^{\sim}u+\lambda u=f$

ThisCompleteStheprOOfof eSSentialm-aCCretiVity of$A+\kappa B$ fOr $\kappa\in\partial\sum$with ${\rm Re}\kappa\geq 0$

.

口

Theorem 3.5. Let $H,$ $A,$ $B,$ $B_{\epsilon},$ $\Sigma$ and

$\gamma$ be the

same as

those in Theorem 3.1 with

$(\gamma 1)-(\gamma 4)$. Let $\alpha_{0}>0$. Assume that $B_{\epsilon}^{1/2}$

is $A^{1/2}$-bounded, with

(3.9) $\alpha_{0}\Vert B_{\epsilon}^{1/2}u\Vert^{2}\leq\cdot\Vert A^{1/2}u\Vert^{2}$, $u\in D(A^{1’ 2})$.

Assume

_further

that $\Sigma$ and

$\gamma$ satisfy $(\gamma 5)_{\alpha_{0}}$ instead

of

$(\gamma 5)_{0}$:

$(\gamma 5)_{\alpha 0}-\alpha_{0}\leq\gamma(0)$

.

Then the following (i) and (ii) hold.

(i) $B$ is _{$(A+\kappa B)$}-bounded

for

$\kappa\in\Sigma^{c}$, with

(3.10) $\Vert Bu\Vert\leq$ dist$(\kappa, \Sigma)^{-1}\Vert(A+\kappa B)u\Vert$, $u\in D(A)\cap D(B)$,

and $\{A+\kappa B;\kappa\in\Sigma^{c}\}$

forms

a holomorphicfamily

of

type (A). In particular,

if

$\gamma(0)<0$,

then $B$ is A-bounded with

(3.11) $\Vert Bu\Vert\leq$ dist$(0, \Sigma)^{-1}\Vert Au\Vert$, _{$u\in D(A)\subset D(B)$}

.

(ii) $A+\kappa B$ is m-accretive in $H$

for

$\kappa\in\Sigma^{c}$ with ${\rm Re}\kappa\geq-\alpha_{0}$ and $A+\kappa B$ is essentially

m-accretive in $H$

for

$\kappa\in\partial\Sigma$ with ${\rm Re}\kappa\geq-\alpha_{0}$. Moreover, $A+\kappa B$ is regularly m-accretive

in $H$

for

$\kappa\in\Sigma^{c}$ with ${\rm Re}\kappa>-\alpha_{0}$, with

(3.12) $|{\rm Im}((A+\kappa B)u, u)|\leq(\tan|\arg(\kappa+\alpha_{0})|){\rm Re}((A+\kappa B)u, u)$, $u\in D(A)\cap D(B)$

.

Proof. (i) The closedness of $A+\kappa B$ for $\kappa\in\Sigma^{c}$ is a consequence of Theorem 3.1. Noting

that $\gamma(0)<0$ implies $0\in\Sigma^{c}$, we see from $(\gamma 4)$ that if _{$\gamma(0)<0$}, then

(3.13) $\Vert B_{\epsilon}u\Vert\leq$ dist$(0, \Sigma)^{-1}\Vert Au\Vert$, $\epsilon>0,$ $u\in D(A)$.

Letting $\epsilon\downarrow 0$ in (3.13) for $u\in D(A)$,

we

obtain (3.11).

(ii) Let $f\in H,$ $\lambda>0$ and $\kappa\in\Sigma^{c}$ with ${\rm Re}\kappa\geq-\alpha_{0}$. Then we consider the equation

(3.14) $Au,$ $+\kappa B_{\epsilon}u_{\epsilon}+\lambda u_{\epsilon}=f$.

In order to prove $R(A+\kappa B+\lambda)=H$ we only have to show that

11

$u_{\epsilon}\Vert,$ $||Au_{\epsilon}||$ and $||B_{\epsilon}u_{\epsilon}||$ are bounded as $\epsilon$ tends to zero. (3.9) implies that $A+\kappa B_{\epsilon}$ is accretive:

${\rm Re}((A+\kappa B_{\epsilon})u, u)=\Vert A^{1\prime 2}u\Vert^{2}+({\rm Re}\kappa)||B_{\epsilon}^{1’ 2}u\Vert^{2}$

$\geq(\alpha_{0}+{\rm Re}\kappa)\Vert B_{\epsilon}^{1/2}u\Vert^{2}$

$\geq 0$.

The accretivity of $A+\kappa B_{\epsilon}$ yields that $\Vert u_{\epsilon}\Vert\leq\lambda^{-1}\Vert f\Vert$

.

$(\gamma 1)-(\gamma 4)$ yield that there exists

$c>0$ such that $\Vert$Au,$\Vert\leq c\Vert f\Vert$ and $\Vert B_{\epsilon}\tau\iota_{\epsilon}\Vert\leq c\Vert f\Vert$. As in the proof of Theorem 3.1, we

obtain $R(A+\kappa B+\lambda)=H$. In particular, if ${\rm Re}\kappa>-\alpha_{0}$, then the numerical range of

4. Proof of Theorem 1.1

In order to prove Theorem 1.1 we need

some

inequalities in the realor complex Hilbert

space $L^{2}(\mathbb{R}^{N})$. We review the following lemma proposed by Ozawa-Sasaki [9].

Lemma 4.1. [9, Theorem 1.1] Let $1\leq p<\infty$.

If

$v\in.L^{p}(\mathbb{R}^{N})$ and $x\cdot\nabla v\in L^{p}(\mathbb{R}^{N})$,

then

(4.1) $\frac{N}{p}\Vert v\Vert\leq\Vert x\cdot\nabla v\Vert$.

Here we give a simple proof of (4.1) when $p=2$.

Proof. Let $v\in L^{2}(\mathbb{R}^{N})$ and $x\cdot\nabla v\in L^{2}(\mathbb{R}^{N})$. Integration by parts gives

(4.2) ${\rm Re}(\uparrow),$$x \cdot\nabla\uparrow))=-\frac{N}{2}\Vert\uparrow)\Vert^{2}$.

Then the Cauchy-Schwarz inequality applies to give (4.1). $\square$

Using two real parameters, we can obtain the following lemma which plays an

impor-tant role to derive

some

inequalities.

Lemma 4.2.

_If

$v\in L^{2}(\mathbb{R}^{N})$ and $|x|^{2}\Delta v\in L^{2}(\mathbb{R}^{N})$, then $|x||\nabla v|\in L^{2}(\mathbb{R}^{N})$ and

(4.3) $0\leq\Vert|x|\nabla v\Vert^{4}+4\Vert x\cdot\nabla v\Vert^{2}\Vert v\Vert^{2}-2N\Vert|x|\nabla v\Vert^{2}\Vert v\Vert^{2}\leq\Vert|x|^{2}\Delta v\Vert^{2}\Vert v\Vert^{2}$

.

Proof. Let $v\in L^{2}(\mathbb{R}^{N})$ with $|x|^{2}\Delta v\in L^{2}(\mathbb{R}^{N})$ and $c_{1},$$c_{2}\in \mathbb{R}$

.

We start with the trivial

inequality

(4.4) $0\leq\Vert|x|^{2}\Delta v+c_{1}x\cdot\nabla v+c_{2}v\Vert^{2}$

$=\Vert|x|^{2}\Delta v\Vert^{2}+c_{1}^{2}\Vert x\cdot\nabla v\Vert^{2}+c_{2}^{2}\Vert v\Vert^{2}$

$+2c_{1}{\rm Re}(x\cdot\nabla v, |x|^{2}\Delta v)+2c_{2}{\rm Re}(|x|^{2}\triangle v, v)+2c_{1}c_{2}{\rm Re}(v, x\cdot\nabla v)$ .

Integration by parts gives

(4.5) ${\rm Re}(x \cdot\nabla v,|x|^{2}\triangle v)=\sum_{j,k=1}^{N}{\rm Re}\int_{\mathbb{R}^{N}}|x|^{2}x_{j}\frac{\partial\tau}{\partial x_{j}}\overline{\frac{\partial^{2_{8)}}}{\partial x_{k}^{2}}}dx$

$=- \sum_{j,k=1}^{N}{\rm Re}\int_{R^{N}}(2x_{j}x_{k}\frac{\partial v}{\partial x_{j}}+|x|^{2}\delta_{jk}\frac{\partial v}{\partial x_{j}}+|x|^{2}x_{j}\frac{\partial^{2}v}{\partial x_{j}\partial x_{k}})\overline{\frac{\partial v}{\partial x_{k}}}dx$

$=-2 \Vert x\cdot\nabla v\Vert^{2}-\Vert|x|\nabla v\Vert^{2}-\frac{1}{2}\sum_{j,k=1}^{N}\int_{\mathbb{R}^{N}}|x|^{2}x_{j}\frac{\partial}{\partial x_{j}}|\frac{\partial v}{\partial x_{k}}|^{2}dx$

(4.6) $(|x|^{2}\triangle\uparrow),$ $v)= \sum_{k=1}^{N}\int_{R^{N}}|x|^{2}\overline{\uparrow)}\frac{\partial^{2}\uparrow)}{\partial x_{k}^{2}}dx$

$=- \sum_{k=1}^{N}\int_{R^{N}}(|x|^{2}\overline{\frac{\partial v}{\partial x_{k}}}+2x_{k}\overline{v})\frac{\partial v}{\partial x_{k}}dx$

$=-\Vert|x|\nabla v\Vert^{2}-2(x\cdot\nabla v, v)$

.

In view of (4.2) and (4.6) we have

(4.7) ${\rm Re}(|x|^{2}\triangle v, v)=-\Vert|x|\nabla\uparrow)\Vert^{2}+N\Vert v\Vert^{2}$

.

Putting (4.2), (4.5) and (4.6) in (4.4), we have

(4.8) $0\leq\Vert|x|^{2}\Delta v\Vert^{2}+(c_{1}^{2}-4c_{1})\Vert x\cdot\nabla v\Vert^{2}+(Nc_{1}-2c_{2})\Vert|x|\nabla\tau’\Vert^{2}$ $+c_{2}(c_{2}+2N-Nc_{1})\Vert v\Vert^{2}$.

Minimizing the right-hand side of (4.8), i.e., setting $c_{1}=2,$ $c_{2}=\Vert|x|\nabla v\Vert^{2}\Vert v\Vert^{2}$ for

$v\neq 0$, we can obtain the second inequality of (4.3). The first inequality of (4.3)

can

be

shown by completing the square as

$( \Vert|x|\nabla v\Vert^{2}-N\Vert v\Vert^{2})^{2}+4\Vert v\Vert^{2}(\Vert x\cdot\nabla v\Vert^{2}-\frac{N^{2}}{4}\Vert v\Vert^{2})$

.

In fact, the nonnegativity of the second term is a consequence of (4.1). $\square$

Lemma 4.3. Let $\epsilon>0$. Then

(4.9) ${\rm Re}(\Delta^{2}u, (|x|^{4}+\epsilon)^{-1}\uparrow 4)\geq-\kappa_{0}(N)\Vert(|x|^{4}+\epsilon)^{-1}u\Vert^{2},$ $u\in H^{4}(\mathbb{R}^{N})$,

(4.10) $\Vert\Delta u\Vert^{2}\geq\alpha_{0}(N)\Vert(|x|^{2}+\epsilon)^{-1}u\Vert^{2},$ $u\in H^{2}(\mathbb{R}^{N}),$ _{$N\geq 5$}

.

Here $\kappa_{0}(N)$ and $\alpha_{0}(N)$

are

defined

as

$\kappa_{0}(N);=\{\begin{array}{ll}112-3(N-2)^{2}, N\leq 8,-\frac{N}{16}(N-8)(N^{2}-16), N\geq 9,\end{array}$

$N^{2}$

$\alpha_{0}(N):=-(N-4)^{2},$ $N\geq 5$.

16

The approximate Rellich inequality (4.10) is already shown in [7, Theorem 6.8] in 1982.

Here we

can

give another proof of (4.10).

Proof. First

we

shall prove (4.9). Put IP:$=(\Delta^{2}u, (|x|^{4}+\epsilon)^{-1}u)$ and $v:=(|x|^{4}+\epsilon)^{\sim 1}u$

for $u\in H^{4}(\mathbb{R}^{N})$

.

Then IP is written as

(4.11) IP $=(\Delta^{2}((|x|^{4}+\epsilon)v), v)$

$=(\Delta((|x|^{4}+\epsilon)v), \triangle v)$

$=(|x|^{4}\Delta v+8|x|^{2}x\cdot\nabla v+4(N+2)|x|^{2}v, \Delta v)+\epsilon\Vert\Delta v\Vert^{2}$ $=(|x|^{2}\triangle v+8x\cdot\nabla v+4(N+2)v, |x|^{2}\Delta v)+\epsilon\Vert\Delta v\Vert^{2}$

.

From (4.5) and (4.6) we have

(4.12) ${\rm Re}$IP $\geq\Vert|x|^{2}\triangle\uparrow’\Vert^{2}-16\Vert x\cdot\nabla v\Vert^{2}-8\Vert|x|\nabla\uparrow)\Vert^{2}+4N(N+2)\Vert v\Vert^{2}$

.

Applying Lemma 4.2 to the first term of the right-hand side of (4.12) multiplied by $\Vert v\Vert^{2}$,

we have

$\Vert v\Vert^{2}{\rm Re}$IP $\geq\Vert|x|\nabla v\Vert^{4}-12\Vert x\cdot\nabla v\Vert^{2}\Vert v\Vert^{2}$

-2(N $+$ 4)$|||$x$|\nabla$v$|$

鴎

$|$v$||$2 _$+$ 4N(N $+$ 2)$||$v$||$

4.

Since $\Vert x\cdot\nabla v\Vert^{2}\leq\Vert|x|\nabla v\Vert^{2}$, it follows that

(4.13) $\Vert v\Vert^{2}{\rm Re}$IP $\geq\Vert|x|\nabla v\Vert^{4}-2(N+10)\Vert|x|\nabla v\Vert^{2}\Vert v\Vert^{2}+4N(N+2)\Vert v\Vert^{4}$

$=[\Vert|x|\nabla v\Vert^{2}-(N+10)\Vert v\Vert^{2}]^{2}-[112-3(N-2)^{2}]\Vert v\Vert^{4}$

.

Hence we obtain ${\rm Re}$IP $\geq-[112-3(N-2)^{2}]\Vert v\Vert^{2}$. In particiilar, if $N\geq 9$, then we see

from Lemma4.1 that

$\Vert$

国 $\nabla v\Vert^{2}-(N+10)\Vert v\Vert^{2}\geq\Vert x\cdot\nabla v\Vert^{2}-(N+10)\Vert v\Vert^{2}$

$\geq(N^{2}\prime 4-N-10)\Vert v\Vert^{2}$

$\geq 0$

.

Applying this inequality to (4.13) implies

$\Vert v\Vert^{2}{\rm Re}$IP $\geq[(\frac{N^{2}}{4}-N-10)\Vert v\Vert^{2}]^{2}-[112-3(N-2)^{2}]\Vert v\Vert^{4}$

$=-[- \frac{N}{16}(N-8)(N^{2}-16)]\Vert v\Vert^{4}$

.

Therefore we obtain ${\rm Re}$IP _{$\geq-\kappa_{0}(N)\Vert v\Vert^{2}$} which is nothing but (4.9).

Next we give a simplified proof of (4.10). Let $v$ $:=(|x|^{2}+\epsilon)^{-1}u$ for $u\in H^{2}(\mathbb{R}^{N})$

.

Then

it follows from (4.2) that

${\rm Re}(-\Delta\uparrow\nu, (|x|^{2}+\epsilon)^{-1}u)={\rm Re}(-\triangle(|x|^{2}v+\epsilon v), v)$ $={\rm Re}(\nabla(|x|^{2}v+\epsilon v), \nabla v)$

$={\rm Re}(|x|^{2}\nabla v+2xv+\epsilon\nabla v, \nabla v)$

$=\Vert|x|\nabla v\Vert^{2}-N\Vert v\Vert^{2}+\epsilon\Vert\nabla v\Vert^{2}$

.

Hence Lemma 4.1 implies

${\rm Re}(-\Delta u, (|x|^{2}+\epsilon)^{-1}\uparrow 4)\geq\Vert x\cdot\nabla v\Vert^{2}-N\Vert v\Vert^{2}$

$\geq\frac{N}{4}(N-4)\Vert v\Vert^{2}$

.

Therefore the SChwarz inequality applieS to give $($410$)$ 口

Proof

of

Theorem 1.1. Let $H$ $:=L^{2}(\mathbb{R}^{N}),$ $A:=\Delta^{2}$ with _$D(A)$ $:=H^{4}(\mathbb{R}^{N})$ and $B$ $:=|x|^{-4}$

with $D(B)$ $:=\{u\in H;|x|^{-4}u\in H\}$

.

Then we see that $B_{\epsilon}=|x|^{-4}(1+\epsilon|x|^{-4})^{-1}=(|x|^{4}+$

$\epsilon)^{-1}$ for $\epsilon>0$

.

Therefore Lemma 4.3 allows us to apply Theorem 2.1 with $k_{0}=\kappa_{0}(N)$ if

5.

Proof

of

Theorem

1.2

In this section we generalize the inequalities obtained in Section 4. To see this

we

propose the generalized discriminant of bi-form in Hilbert spaces.

Lemma 5.1. Let $X$ be a complex Hilbert space with inner product $(\cdot,$$\cdot)x$ and

norm

11

$\Vert_{X}$

.

Let $\varphi\in X,$ $c\in \mathbb{R}$ and let

M.be

a

selfadjoint operator in X. Assume that

for

every

$\zeta\in D(M)$,

(5.1) $(M\zeta, \zeta)_{X}+2{\rm Re}(\varphi, \zeta)_{X}+c\geq 0$.

Then $M$ is nonnegative and

(5.2) $S_{1}11\epsilon>0p((M+\epsilon)^{-1}\varphi, \varphi)_{X}\leq c$.

In particular,

_if

$M$ is positive, then

(5.3) $(M^{-1}\varphi, \varphi)_{X}\leq c$.

Proof. First we shall show that $M$ is nonnegative. Considering $\zeta’\Vert\zeta\Vert_{X}$ instead of $\zeta$, it

suffices to show that $(M\zeta, \zeta)_{X}\geq 0$ for $\zeta\in D(M)$ with $\Vert\zeta$

Il

$x=1$. Let $t\in \mathbb{R}$ with $t\neq 0$.

Then it follows from (5.1) with $\zeta$ replaced with $t\zeta$ that

$0\leq t^{2}(M\zeta, \zeta)_{X}+2t{\rm Re}(\varphi, \zeta)_{X}+c$

$\leq t^{2}(M\zeta, \zeta)_{X}+2|t|\Vert\varphi\Vert_{X}+c$

.

This is equivalent to

$-2|t|^{-1}\Vert\varphi\Vert_{X}-ct^{-2}\leq(M\zeta, \zeta)_{X}$.

Letting $|t|arrow\infty$ yields that $(M\zeta, \zeta)_{X}\geq 0$. Next we shall prove (5.2). Let $M_{\epsilon}$ $:=M+\epsilon$.

Since $M$ is nonnegative selfadjoint in $X$, we see that $M_{\epsilon}^{-1}$ is well-defined as a bounded

symmetric operator with $\Vert A\prime f_{\epsilon}^{-1}\zeta\Vert_{X}\leq\epsilon^{-1}\Vert\zeta\Vert_{X}$. Then (5.1) implies that

$0\leq(M_{\epsilon}\zeta, \zeta)_{X}+2{\rm Re}(\varphi, \zeta)_{X}+c$

$=(M_{\epsilon}(\zeta+M_{\epsilon}^{-1}\varphi), \zeta+M_{\epsilon}^{-1}\varphi)_{X}-(M_{\epsilon}^{-1}\varphi, \varphi)_{X}+c$.

Taking $\zeta=-M_{\epsilon}^{-1}\varphi$, we see that $(M_{\epsilon}^{-1}\varphi, \varphi)_{X}\leq c$ for $\epsilon>0$. Therefore we obtain (5.2).

In particular, if $M$ is positive, then we can take $\epsilon=0$. $\square$

Using two complex parameters, we can obtain the following lemma which is

a

strict

version of Lemma 4.1

Lemma 5.2.

_If

$v\in L^{2}(\mathbb{R}^{N})$ and $x\cdot\nabla v\in L^{2}(\mathbb{R}^{N})$, then

(5.4) $|{\rm Im}(v, x \cdot\nabla v)|^{2}\leq\Vert v\Vert^{2}(\Vert x\cdot\nabla v\Vert^{2}-\frac{N^{2}}{4}\Vert v\Vert^{2})$.

Proof. Let $v\in L^{2}(\mathbb{R}^{N})$ with $x\cdot\nabla v\in L^{2}(\mathbb{R}^{N})$. From the Schwarz inequality we have

(5.5) $|(v, x\cdot\nabla v)|^{2}\leq\Vert v\Vert^{2}\Vert x\cdot\nabla v\Vert^{2}$

.

If $X$ $:=\mathbb{C}^{2}$, then Lemma 5.1 is regarded a.s a

two-complex-parameter technique to

derive a new inequality.

Corollary 5.3. Let $M$ be

a

Hermite matnx

on

$\mathbb{C}^{2}$ :

$M=(\begin{array}{ll}b \gamma\overline{\gamma}a \end{array})$

where $a,$ $b\in \mathbb{R}$ and$\gamma\in \mathbb{C}$

.

Assume that there

are

$\varphi:={}^{t}(\overline{\alpha},$$\beta)\in \mathbb{C}^{2}$ and$c\in \mathbb{R}_{l}$ satisfying

(5.1). Then it

_{follows from}

(5.2) that

$a|\alpha|^{2}+b|\beta|^{2}-2{\rm Re}(\alpha\beta\gamma)\leq c(ab-|\gamma|^{2})$

.

Setting $\alpha:=\alpha_{1}+i\alpha_{2},$ $\beta:=\beta_{1}+i\beta_{2}$, $\gamma$ $:=\gamma_{1}+i\gamma_{2}$,

one

has

(5.6) $a\alpha_{2}^{2}+b\beta_{2}^{2}+c\gamma_{2}^{2}+2(\alpha_{1}\beta_{2}\gamma_{2}+\alpha_{2}\beta_{1}\gamma_{2}+\alpha_{2}\beta_{2}\gamma_{1})$

$\leq abc+2\alpha_{1}\beta_{1}\gamma_{1}-(a\alpha_{1}^{2}+l_{J}\beta_{1}^{2}+c\gamma_{1}^{2})$

.

The following lemma together with Lemma 5.2 give a strict version of Lemma 4.2.

Lemma 5.4.

_If

$v\in L^{2}(\mathbb{R}^{N})$ and $|x|^{2}\Delta v\in L^{2}(\mathbb{R}^{N})$, then $|x||\nabla v|\in L^{2}(\mathbb{R}^{N})$ and

(5.7) $[\Vert v\Vert^{2}{\rm Im}(x\cdot\nabla v, |x|^{2}\Delta v)-\Vert|x|\nabla v\Vert^{2}{\rm Im}(v, x\cdot\nabla v)]^{2}$

$\leq[\Vert v\Vert^{2}\Vert x\cdot\nabla v\Vert^{2}-\frac{N^{2}}{4}\Vert v\Vert^{4}-|{\rm Im}(v, x\cdot\nabla v)|^{2}]$

$\cross[\Vert|x|^{2}\Delta v\Vert^{2}\Vert v\Vert^{2}+2N\Vert|x|\nabla v\Vert^{2}\Vert v\Vert^{2}-\Vert|x|\nabla v\Vert^{4}-4\Vert x\cdot\nabla v\Vert^{2}\Vert v\Vert^{2}]$

.

Proof. Let $v\in L^{2}(\mathbb{R}^{N})$ with $|x|^{2}\Delta v\in L^{2}(\mathbb{R}^{N})$. Then for $\zeta={}^{t}(\zeta_{1},$ $\zeta_{2})\in \mathbb{C}^{2}$ we have

an

inequality of the form (5.1):

$0\leq\Vert|x|^{2}\Delta v+\zeta_{1}(x\cdot\nabla)v+\zeta_{2}v\Vert^{2}$

$=(M\zeta, \zeta)_{\mathbb{C}^{2}}+2{\rm Re}(\varphi, \zeta)_{\mathbb{C}^{2}}+c$,

where $\varphi={}^{t}(\overline{\alpha},$$\beta)$ $:=(\overline{((x\cdot\nabla)v,|x|^{2}\Delta v)}, (|x|^{2}\Delta v, v)),$ _$c$ $:=\Vert|x|^{2}\Delta v\Vert^{2}$ and

$M=(\begin{array}{ll}b \gamma\overline{\gamma}a \end{array});=$ $( \frac{||(x\cdot\nabla)v||^{2}}{(v,(x\cdot\nabla)v)}$ $(v, (x\cdot\nabla)v)||v\Vert^{2})$

.

Thus we obtain (5.6)

as

a consequence of Corollary 5.3. Now it is easy to

see

from (4.2),

(4.5) and (4.6) that

(5.8) $\alpha_{1}={\rm Re}\alpha=\frac{N}{2}\tilde{b}-2b$,

(5.9) $\beta_{1}={\rm Re}\beta=Na-\tilde{b}$,

where$\tilde{b}:=\Vert|x|\nabla v\Vert^{2}$. It follows from $(5.8)-(5.10)$ that the right-hand side of (5.6) equals

$(b-(N^{2}/4)a)(ac+2Na\tilde{b}-\tilde{b}^{2}-4ab)$.

Multiplying (5.6) by $a$ and using the equality $\beta_{2}=2\gamma_{2}$, we have

(5.11) $a^{2}\alpha_{2}^{2}+2a(\beta_{1}+2\gamma_{1})\alpha_{2}\gamma_{2}+a(4\alpha_{1}+4b+c)\gamma_{2}^{2}$ $\leq a(b-(N^{2}/4)a)(ac+2Na\tilde{b}-\tilde{b}^{2}-4ab)$

.

We see from $(5.8)-(5.10)$ that the left-hand side of (5.11) equals

$(a\alpha_{2}-\tilde{b}\gamma_{2})^{2}+(ac+.2Na\tilde{b}-\tilde{b}^{2}-4ab)\gamma_{2}^{2}$,

which implies that

$(a\alpha_{2}-\tilde{l_{J}}\gamma_{2})^{2}\leq(ab-(N^{2}4)a^{2}-\gamma_{2}^{2})(ac+2Na\tilde{b}-\tilde{l_{J^{2}}}-4ab)$

.

This proves (5.7). $\square$

Lemma 5.5. Let $u\in H^{4}(\mathbb{R}^{N})$ and $\epsilon>0$. Let $k_{1}$ and $k_{2}$ be constants

defined

as $k_{1}:=112-3(N-2)^{2}$,

$k_{2}:=- \frac{N}{16}(N-8)(N^{2}-16),$ $N\geq 9$.

Put IP:$=(\Delta^{2}u, (|x|^{4}+\epsilon)^{-1}u)$ and _$a:=||$$(|x|^{4}+\epsilon)^{-1}u\Vert^{2}$

.

Then

(5.12) $({\rm Im}$IP$)2 \leq 64\sqrt{a}(\sqrt{{\rm Re} IP+k_{1}a}+(10+N-\frac{N^{2}}{4})\sqrt{a})(\sqrt{{\rm Re} IP+k_{1}a}+8\sqrt{a})^{2}$

.

If

$N\geq 9$, then it is equivalent to

(5.13) $({\rm Im}$IP$)2 \leq\frac{64\sqrt{a}({\rm Re} IP+k_{2}a)(\sqrt{{\rm Re} IP+k_{1}a}+8\sqrt{a})^{2}}{\sqrt{{\rm Re} IP+k_{1}a}+(\frac{N^{2}}{4}-N-10)\sqrt{a}}$

.

Proof. Let $u\in H^{4}(\mathbb{R}^{N})$ and $\epsilon>0$

.

Put $v:=(|x|^{4}+\epsilon)^{-1}u$

.

Using the same notations ais

in the proof of Lemma 5.4, we see that (5.7) is written as

(5.14) $L:= \frac{(a\alpha_{2}-\tilde{b}\gamma_{2})^{2}}{ab-(N^{2}’ 4)a^{2}-\gamma_{2}^{2}}\leq ac+2Na\tilde{b}-\tilde{b}^{2}-4ab=:R$.

Here we note (4. 11) that

IP $=\Vert|x|^{2}\Delta v\Vert^{2}+8((x\cdot\nabla)v, |x|^{2}\Delta v)+4(N+2)(v, |x|^{2}\Delta v)+\epsilon\Vert\Delta v\Vert^{2}$.

Since $\beta_{2}=2\gamma_{2}$, it follows that

(5.15) $c=\Vert|x|^{2}\triangle v\Vert^{2}\leq{\rm Re} IP+16b+8\tilde{b}-4N(N+2)a$,

Applying (5.16) to $L$ yields $L= \frac{(\frac{a}{8}In1IP+((N+2)a-\tilde{b})\gamma_{2})^{2}}{a(b-(N^{2}/4)a)-\gamma_{2}^{2}}=\frac{(c_{1}\gamma_{2}+c_{2})^{2}}{c_{0}-\gamma_{2}^{2}}$ , where (5.17) $c_{0}:=a(b-(N^{2}4)a)\geq\gamma_{2}^{2}$, (5.18) $c_{1}:=(N+2)a-\tilde{J_{J}}$, (5.19) $c_{2}:= \frac{a}{8}In1$IP;

note

_that.

the inequality in (5.17) is nothing but (5.4). Since the quadratic equation

$L(c_{0}-t^{2})=(c_{1}t+c_{2})^{2}$ has a real root $t=\gamma_{2}$, the discriminant is nonnegative:

(5.20) $L(c_{0}L+c_{0}c_{1}^{2}-c_{2}^{2})\geq 0$

.

It is clear that $L\geq 0$

.

If $L>0$, then (5.20) yields

(5.21) $L\geq(c_{2}^{2}c_{0})-c_{1}^{2}$

.

If$L=0$, then $\gamma_{2}=-c_{2}/c_{1}$ and hence (5.17) yields that $0\geq(c_{2}^{2}/c_{0})-c_{1}^{2}$

.

This

means

that

(5.21) holds for $L\geq 0$

.

Hence it follows from _{$(5.17)-(5.19)$} and (5.21) that

(5.22) $L \geq\frac{a|In1IP|^{2}}{64(b-(N^{2}’ 4)a)}-(\tilde{b}-(N+2)a)^{2}$

.

On the other hand, since $b\leq\tilde{b},$

$(5.14)$ and (5.15) yields

$R\leq a{\rm Re} IP+12ab+2(N+4)a\tilde{b}-\overline{b}^{2}-4N(N+2)a^{2}$

(5.23) $\leq a(k_{1}a+{\rm Re} IP)$ $-(\tilde{b}-(N+10)a)^{2}$,

where $k_{1}$ $:=(N+10)^{2}-4N(N+2)=112-3(N-2)^{2}$. Since $L\leq R$, it follows from

(5.22) and (5.23) that

(5.24) $\frac{a|In1IP|^{2}}{64(b-N^{2}a/4)}-(\tilde{b}-(N+2)a)^{2}\leq a(k_{1}a+{\rm Re} IP)$ $-(\tilde{b}-(N+10)a)^{2}$

.

Therefore

we

obtain

(5.25) $\frac{|{\rm Im} IP|^{2}}{64(b-(N^{2}’ 4)a)}-16(\tilde{b}-(N+6)a)\leq k_{1}a+{\rm Re}$IP $=:K$

.

Now

we

see from (5.23) that

$(\tilde{J_{J}}-(N+10)a)^{2}\leq R+(\tilde{l_{J}}-(N+10)a)^{2}\leq aK$

and hence

Applying (5.26) to (5.25), we obtain

$\frac{|In\iota IP|^{2}}{64\sqrt{a}[\sqrt{K}-((N^{2}’ 4)-N-10)\sqrt{a}]}\leq K+16(\sqrt{aK}+4a)=(\sqrt{K}+8\sqrt{a})^{2}$ .

This proves (5.12). Next note that $N^{2}/4-N-10\geq 0$ for $N\geq 9$. To obtain (5.13),

we

have only to use the equality

$\sqrt{K}-((N^{2}/4)-N-10)\sqrt{a}=\frac{k_{2}a+{\rm Re} IP}{\sqrt{K}+((N^{2}/4)-N-10)\sqrt{a}}$

where $k_{2}=-N(N-8)(N^{2}-16)/16$. $\square$

Proof

of

Theorem 1.2. Let $H$ $:=L^{2}(\mathbb{R}^{N}),$ $A$ $:=\Delta^{2}$ with _$D(A)$ $:=H^{4}(\mathbb{R}^{N})$ and $B$ $:=|x|^{-4}$

with $D(B)$ $:=\{u\in H;|x|^{-4}u\in H\}$. For $u\in D(A)$ and $\epsilon>0$ take $\uparrow J;=B_{\epsilon}u=$ $(|x|^{4}+\epsilon)^{-1}u$ with $\sqrt{a};=\Vert v\Vert=1$. Then set

$\xi+i\eta:=-$IP $=-(Au, B_{\epsilon}u)$.

If $N\leq 8$, then $\xi\leq k_{1}$ $:=112-3(N-2)^{2}$. In fact, we see from (4.9) that

$-\xi={\rm Re}$IP $\geq-[112-3(N-2)^{2}]$ for $v\in H$ with $\Vert v\Vert=1$

.

Thus (5.12) $($with ${\rm Re}$IP $=-\xi,$${\rm Im}$IP

$=-\eta,$$a=1)$ allows lls to apply Theorem 3.1 with

$\Sigma;=\{\xi+i\eta\in \mathbb{C};\xi\leq k_{1}, \eta^{2}\leq\varphi_{N}(\xi)\}$,

$\gamma(\eta)+i\eta\in\partial\Sigma(\Rightarrow\gamma(0)=k_{1}>0)$,

where

$\varphi_{N}(\xi):=64[\sqrt{k_{1}-\xi}+(10+N-(N^{2}/4))](\sqrt{k_{1}-\xi}+8)^{2}$, $\xi\leq k_{1}$.

In more detail $\gamma$ is given by

$\gamma(\eta):=\{\begin{array}{ll}k_{1}, |\eta|\leq\eta_{N},\varphi_{N}^{-1}(\eta^{2})\Leftrightarrow\eta^{2}=\varphi_{N}(\gamma(\eta)), |\eta|\geq 7lN,\end{array}$

where $\eta_{N}$ $:=\sqrt{\varphi_{N}(k_{1})}=\sqrt{\min\varphi_{N}}=64\sqrt{10+N-(N^{2}\prime 4)}$

.

In particular, if $N\geq 5$,

then the Rellich inequality (4.10)

$(N4)(N-4)\Vert(|x|^{2}+\epsilon)^{-1}u\Vert\leq\Vert\Delta u\Vert$, $u\in H^{2}(\mathbb{R}^{N})$

applies to give (3.9) with $\alpha_{0}$ $:=(N^{2}/16)(N-4)^{2}$. In fact, it follows for every $u\in$

$D(A)\cap D(B)$ that $u\in D(A^{1/2})\subset D(B^{1/2})$ and

$\alpha_{0}((|x|^{4}+\epsilon)^{-1}\tau\iota, u)\leq\alpha_{0}(|x|^{-4}u, u)=\alpha_{0}\Vert|x|^{-2}u\Vert^{2}\leq\Vert\Delta u\Vert^{2}=(\Delta^{2}u, u)$

.

If $N\geq 9$, then we have _{$\xi\leq k_{2};=-(N/16)(N-8)(N^{2}-16)$}. In fact, it follows from

(4.9) that

$-\xi={\rm Re}$IP $\geq(N/16)(N-8)(N^{2}-16)$ for $v\in H$ with $\Vert v\Vert=1$.

Thus (5.13) allows us to apply Theorem 3.5 with $\alpha_{0}:=(N^{2}16)\cdot(N-4)^{2}$ and

$\Sigma;=\{\xi+i\eta\in \mathbb{C};\xi\leq k_{2}, \eta^{2}\leq\varphi_{N}(\xi)\}$,

$\gamma(\eta)+i\eta\in\partial\Sigma(\Rightarrow-\alpha_{0}<\gamma(0)=k_{2}<0)$

.

where

$\varphi_{N}(\xi):=\frac{64(k_{2}-\xi)(\sqrt{k_{1}-\xi}+8)^{2}}{\sqrt{k_{1}-\xi}+((N^{2}/4)-N-10)}.$, $\xi\leq k_{2}$.

$\gamma$ is given by $\gamma(\eta)$ $:=\varphi_{N}^{-1}(\eta^{2})$. This completes the proof of Theorem 1.2. $\square$

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